arxiv: 2604.08892 · v1 · submitted 2026-04-10 · 💻 cs.CG

Recognition: unknown

Parametric Shortest Paths in a Linearly Interpolated Graph

Jacob Sriraman , Eli Barton , Brittany Terese Fasy , David L. Millman , Brendan Mumey , Nate Rengo , Braeden Sopp , Vasishta Tumuluri , Binhai Zhu

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:02 UTC · model grok-4.3

classification 💻 cs.CG

keywords parametric shortest pathslinear interpolationdata structuresgraph algorithmsshortest path queriesdirected graphsinterpolated graphsinterval partitioning

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The pith

In graphs whose edge weights interpolate linearly between two fixed graphs, all distinct shortest paths over intervals of the parameter can be precomputed for logarithmic query time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an algorithm to find every distinct shortest path that is optimal over some nontrivial interval of the interpolation parameter λ between 0 and 1. It builds a compact data structure recording each such path together with its exact maximal interval, at a total cost of Θ(k |E| log |V|) where k is the number of these distinct paths. A reader would care because many real networks have edge costs that change continuously between known states; the structure turns a family of shortest-path problems into one preprocessing pass plus fast lookups instead of repeated full recomputations.

Core claim

The authors present an algorithm that computes all distinct parametric shortest paths in the interpolated graph family (1−λ)G0 + λG1. Each path is optimal over a maximal subinterval of [0,1] and corresponds to a linear cost function in that interval. The algorithm constructs a data structure in Θ(k |E| log |V|) time, after which a shortest-path query for any given λ can be answered in Θ(log k) time.

What carries the argument

The enumeration of maximal λ-intervals on which a single path remains shortest, found by comparing the linear cost functions of candidate paths across the parameter range.

Load-bearing premise

Edge weights stay positive in both endpoint graphs so that shortest paths remain well-defined for every interpolation value between them.

What would settle it

A counterexample pair of small graphs in which the computed structure either reports a path that is not shortest for some λ inside one of its claimed intervals or misses an interval where a different path becomes optimal.

Figures

Figures reproduced from arXiv: 2604.08892 by Binhai Zhu, Braeden Sopp, Brendan Mumey, Brittany Terese Fasy, David L. Millman, Eli Barton, Jacob Sriraman, Nate Rengo, Vasishta Tumuluri.

**Figure 2.** Figure 2: Snapshots of the algorithm before, during, and after it runs. The blue [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

read the original abstract

We consider the parametric shortest paths problem in a linearly interpolated graph. Given two positively-weighted directed graphs $G_0=(V,E,\omega_0)$ and $G_1=(V,E,\omega_1),$ the linearly interpolated graph is the family of graphs $(1-\lambda)G_0+\lambda G_1$, parameterized by $\lambda\in [0,1]$. The problem is to compute all distinct parametric shortest paths. We compute a data structure in $\Theta(k|E|\log |V|)$ time, where~$k$ is the number of distinct parametric shortest paths over all~$\lambda\in [0,1]$ that exist for a nontrivial interval of parameters, each corresponding to a linear function in a maximal sub-interval of $[0,1]$. Using this data structure, a shortest path query takes~$\Theta(\log k)$ time.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an output-sensitive data structure for shortest paths under linear interpolation of edge weights between two fixed positive-weight graphs.

read the letter

The main takeaway is a preprocessing scheme that builds a data structure in Θ(k |E| log |V|) time for the family of graphs (1-λ)G0 + λG1, where k counts the maximal intervals on which a single path remains shortest, then answers any λ-query in Θ(log k) time. This is a direct algorithmic result for the interpolation setting rather than a reduction from earlier parametric shortest-path work. The positive-weight assumption on both input graphs keeps all interpolated instances non-negative, so standard Dijkstra applies at each step without negative cycles or undefined behavior. The construction tracks the breakpoints where the shortest path changes as λ sweeps [0,1] and stores the corresponding linear functions on each interval. That approach is clean and matches the output-sensitive complexity they claim. The shared vertex and edge sets between G0 and G1 simplify the setup, which is reasonable for the problem they pose. One minor limitation is that k can still reach Θ(|V|!) in contrived cases even though the algorithm only pays for the actual number of changes; the paper does not claim sublinear k in the worst case. The work stays within directed graphs, so extending to undirected inputs would need extra handling. This is a targeted contribution in computational geometry for parameter-dependent networks. Readers working on data structures for varying weights or geometric optimization will find the concrete bounds and query time useful. The reasoning is straightforward and the result is falsifiable via implementation. It deserves a serious referee because it supplies a new, explicitly bounded data structure for a well-defined parametric problem.

Referee Report

0 major / 2 minor

Summary. The paper considers the parametric shortest paths problem in a linearly interpolated graph given by two positively-weighted directed graphs G0 and G1. It computes a data structure in Θ(k|E|log |V|) time, where k is the number of distinct parametric shortest paths that exist for nontrivial intervals of λ in [0,1], each corresponding to a linear function on a maximal sub-interval. This data structure allows shortest path queries for any λ to be answered in Θ(log k) time.

Significance. If the result holds, it provides an efficient output-sensitive method for parametric shortest path queries, which is valuable in applications involving parameter-dependent networks. The use of positive weights ensures the applicability of standard algorithms without negative cycles. The output-sensitive complexity is a strength, as it depends on the actual number of distinct paths rather than a worst-case bound.

minor comments (2)

[Abstract] The abstract states the preprocessing time bound explicitly but does not mention the space complexity of the resulting data structure; this should be added for completeness.
A brief discussion or example early in the paper illustrating how linear interpolation preserves positivity and avoids negative cycles would strengthen the presentation of the model assumptions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. We are pleased that the output-sensitive complexity and its potential applications are recognized as strengths.

Circularity Check

0 steps flagged

No circularity: standard output-sensitive algorithmic construction

full rationale

The paper presents a constructive algorithm whose running time is stated directly as Θ(k|E|log |V|), where k counts the number of maximal intervals on which a single path remains shortest. This is an output-sensitive bound derived from standard shortest-path primitives (Dijkstra on non-negative weights) applied across breakpoints; k is defined as the size of the output, not fitted or presupposed. No self-citation is load-bearing for the complexity claim, no ansatz is smuggled, and the derivation does not reduce any equation to itself by construction. The positive-weight assumption on G0 and G1 is external and ensures the interpolated graphs remain valid inputs for the same primitives.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard domain assumptions from graph algorithms; no free parameters or invented entities are introduced.

axioms (1)

domain assumption The input graphs are directed with positive edge weights.
Stated in the abstract to guarantee well-defined shortest paths under linear interpolation.

pith-pipeline@v0.9.0 · 5493 in / 1216 out tokens · 90592 ms · 2026-05-10T17:02:29.859936+00:00 · methodology

discussion (0)

Reference graph

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